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Import Data

Import the AVISO-data from the file 'Agulhas_AVISO.mat' stored in the folder 'Data'.

Data/Parameters for Dynamical System

Spatio-Temporal Domain of Dynamical System

Velocity Interpolation

In order to evaluate the velocity field at arbitrary locations and times, we must interpolate the discrete velocity data. The interpolation with respect to time is always linear. The interpolation with respect to space can be chosen to be "cubic" or "linear". In order to favour a smooth velocity field, we interpolate the velocity field in space using a cubic interpolant.

Polar Rotation Angle (PRA)

Next, we compute the PRA over the meshgrid over the given time-interval. We iterate over all initial conditions and first calculate the gradient of the flow map using an auxiliary grid. 'aux_grid' specifies the ratio between the auxiliary grid and the original meshgrid. This parameter is generally chosen to be between $ [\dfrac{1}{10}, \dfrac{1}{100}] $.

The Polar Rotation Angle (PRA) is then computed from the eigenvectors $ \xi_i\eta_i, \eta_i $ (with i = 1, 2) of the right\left Cauchy-Green strain tensor $ C_{t_0}^{t_N}(\mathbf{x}_0) $:

\begin{equation} \mathrm{PRA}_{t_0}^{t_N}(\mathbf{x}_0) = \langle \xi_1(\mathbf{x}_0;t_0, t_N), \eta_1(\mathbf{x}_0;t_0, t_N) \rangle = \langle \xi_2(\mathbf{x}_0;t_0, t_N), \eta_2(\mathbf{x}_0;t_0, t_N) \rangle \end{equation}

As the maximum eigenvalue is less sensitive with respect to numerical errors, it is recommended to use the dominant eigenvectors $ \xi_2, \eta_2 $ in order to compute $ \mathrm{PRA}_{t_0}^{t_N}(\mathbf{x}_0) $.